Quasi-maximum Likelihood Inference for Linear Double Autoregressive Models

Abstract

This paper investigates the quasi-maximum likelihood inference including\nestimation, model selection and diagnostic checking for linear double\nautoregressive (DAR) models, where all asymptotic properties are established\nunder only fractional moment of the observed process. We propose a Gaussian\nquasi-maximum likelihood estimator (G-QMLE) and an exponential quasi-maximum\nlikelihood estimator (E-QMLE) for the linear DAR model, and establish the\nconsistency and asymptotic normality for both estimators. Based on the G-QMLE\nand E-QMLE, two Bayesian information criteria are proposed for model selection,\nand two mixed portmanteau tests are constructed to check the adequacy of fitted\nmodels. Moreover, we compare the proposed G-QMLE and E-QMLE with the existing\ndoubly weighted quantile regression estimator in terms of the asymptotic\nefficiency and numerical performance. Simulation studies illustrate the\nfinite-sample performance of the proposed inference tools, and a real example\non the Bitcoin return series shows the usefulness of the proposed inference\ntools.\n